Sphere

class pychemengg.heattransfer.steadystate.Sphere(inner_radius=None, outer_radius=None, thermalconductivity=None)

Bases: object

Models a spherical object

Parameters

inner_radius: `int or float`

Inner radius of spherical object

outer_radius: `int or float`

Outer radius of spherical object

thermalconductivityint or float

Thermal conductivity of spherical object

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss
>>> shellLead = ss.Sphere(inner_radius=0.25, outer_radius=0.30, thermalconductivity=35.3)
# This will create an instance of 'Sphere' with a name 'shellLead'  

Attributes

inner_radius: `int or float`

Inner radius of spherical object

outer_radius: `int or float`

Outer radius of spherical object

thermalconductivityint or float

Thermal conductivity of spherical object

__init__(inner_radius=None, outer_radius=None, thermalconductivity=None)

Methods

__init__([inner_radius, outer_radius, …])
area([radius])	Computes surface area of a sphere
heatrateof_cond([dT])	Computes heat rate of conduction for a spherical object
heatrateof_conv([heattransfercoefficient, …])	Computes heat rate of convection for a spherical object
heatrateof_rad([radius, T_infinity, …])	Computes heat rate of radiation for a spherical object
resistanceof_cond()	Computes resistance of conduction for a spherical object
resistanceof_conv([heattransfercoefficient, …])	Computes resistance of convection for a spherical object
resistanceof_fouling([foulingfactor, radius])	Computes resistance of fouling for a spherical object
volume([radius])	Calculates volume of a sphere

area(radius=None)

Computes surface area of a sphere

Parameters

radiusint or float

Radius at which surface area is to be computed

Returns

areaint or float

Surface area of sphere

Notes

Surface area of sphere is computed using:

\[A = 4 \pi r^2 \]

where:

r = radius

A = surface area

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss

>>> shellLead = ss.Sphere(inner_radius=0.25, outer_radius=0.30, thermalconductivity=35.3)
# This will create an instance of 'Sphere' with a name 'shellLead'

>>> shellLead.area(radius=shellLead.outer_radius)
1.1309733552923256
# This computes surface area @ outer radius

>>> shellLead.area(radius=shellLead.inner_radius)
0.7853981633974483
# This computes surface area @ inner radius

>>> shellLead.area(radius=0.25)
0.7853981633974483
# This will also compute surface area @ inner radius, which is = 0.25
# Here the radius is entered as a number directly, rather than as an attribute.

heatrateof_cond(dT=None)

Computes heat rate of conduction for a spherical object

Parameters

dTint or float

Temperature difference between two surfaces of the sphere

Returns

heatrateint or float

Rate of heat transfer by conduction

Notes

The following formula is used:

\[Q (heatrate) = \frac{\Delta T}{R_{conduction}} \]

where:

\(\Delta T\) = temperature difference

\(R_{conduction}\) = conduction resistance given by

\(R_{conduction} = \cfrac{(r_o - r_i)}{4\pi k r_o r_i}\)

\(r_o\) = outer radius of sphere

\(r_i\) = inner radius of sphere

k = thermal conductivity

References

[1] Yunus A. Cengel and Afshin J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss
>>> shellSS = ss.Sphere(inner_radius=.30, outer_radius=.31, thermalconductivity=15.1)
>>> shellSS.heatrateof_cond(dT=75)
132352.15690308428

heatrateof_conv(heattransfercoefficient=None, radius=None, dT=None)

Computes heat rate of convection for a spherical object

Parameters

heattransfercoefficientint or float

Heat transfer coefficient `h` for the spherical surface

radiusint or float

Radius of sphere where convective heat transfer rate is to be computed

dTint or float

Temperature difference between spherical surface and surrounding fluid

Returns

heatrateint or float

Rate of heat transfer by convection

Notes

Heat rate of convection is calculated using the Newton’s Law

\[Q (heatrate) = h A \Delta T \]

where:

h = heat transfer coefficient

A = area of heat transfer

\(\Delta T\) = temperature difference

References

[1] Yunus A. Cengel and Afshin J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss
>>> shellSS = ss.Sphere(inner_radius=.30, outer_radius=.31, thermalconductivity=15.1)
>>> shellSS.heatrateof_conv(heattransfercoefficient=500, radius=shellSS.outer_radius, dT=25)
15095.352700498957

heatrateof_rad(radius=None, T_infinity=None, T_surface=None, emissivity=None)

Computes heat rate of radiation for a spherical object

Parameters

radiusint or float

Radius of sphere where radiation heat transfer rate is to be computed

T_infinityint or float

Temperature of surroundings in absolute temperature units

T_surfaceint or float

Temperature of sphere’s surface in absolute temperature units

emissivityint or float

Emissivity of the sphere

Returns

heatrateint or float (returns a positive value)

Rate of heat transfer by radiation

Notes

Heat rate of radiation is calculated using the Stefan-Boltzmann law

\[Q (heatrate) = \sigma \epsilon A (T_{infinity}^4 - T_{surface}^4) \]

where:

\(\sigma\) = Stefan-Boltzmann constant

\(\epsilon\) = emissivity of object

A = area of heat transfer

\(T_{infinity}^4\) = absolute temperature of surroundings

\(T_{surface}^4\) = absolute temperature of surface

References

[1] Yunus A. Cengel and Afshin J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss
>>> shellSS = ss.Sphere(inner_radius=.30, outer_radius=.31, thermalconductivity=15.1)
>>> shellSS.heatrateof_rad(radius=shellSS.outer_radius, T_infinity=30+273, T_surface=90.6+273, emissivity=0.95)
588.6831572504626

resistanceof_cond()

Computes resistance of conduction for a spherical object

Parameters

`None_required`‘None’

Uses attributes defined during instance creation

Returns

resistanceint or float

Conduction resistance

Notes

The following formula is used:

\[R_{conduction} = \cfrac{(r_o - r_i)}{4\pi k r_o r_i}\]

where:

\(r_o\) = outer radius of sphere

\(r_i\) = inner radius of sphere

k = thermal conductivity

\(R_{conduction}\) = conduction resistance

References

[1] Yunus A. Cengel and Afshin J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss
>>> shellSS = ss.Sphere(inner_radius=.30, outer_radius=.31, thermalconductivity=15.1)
>>> shellSS.resistanceof_cond()
0.0005666700245385441

resistanceof_conv(heattransfercoefficient=None, radius=None)

Computes resistance of convection for a spherical object

Parameters

heattransfercoefficientint or float

Heat transfer coefficient `h` for the spherical surface

radiusint or float

Radius of sphere where convective heat transfer is to be computed

Returns

resistanceint or float

Convection resistance

Notes

The following formula is used:

\[R_{convection} = \frac{1}{hA} \]

where:

h = heat transfer coefficient

A = area of heat transfer

\(R_{convection}\) = convection resistance

References

[1] Yunus A. Cengel and Afshin J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss
>>> shellSS = ss.Sphere(inner_radius=.30, outer_radius=.31, thermalconductivity=15.1)
>>> shellSS.resistanceof_conv(heattransfercoefficient=500, radius=shellSS.outer_radius)
0.0016561388459094206

resistanceof_fouling(foulingfactor=None, radius=None)

Computes resistance of fouling for a spherical object

Parameters

foulingfactorint or float

Fouling factor \(R_f\) for the spherical surface

typical units are \(m^2\) K/W

Returns

resistanceint or float

Fouling resistance

Notes

The following formula is used:

\[R_{fouling} = \frac{R_f}{A} \]

where:

\(R_f\) = fouling factor

A = fouled area of heat transfer

\(R_{fouling}\) = fouling resistance

References

[1] Yunus A. Cengel and Afshin J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss
>>> shellSS = ss.Sphere(inner_radius=.30, outer_radius=.31, thermalconductivity=15.1)
>>> shellSS.resistanceof_fouling(foulingfactor=0.0007, radius=shellSS.outer_radius)
0.0005796485960682973

volume(radius=None)

Calculates volume of a sphere

Parameters

radiusint or float

Radius at which volume is to be computed

Returns

volumeint or float

Volume of sphere

Notes

Volume of sphere is computed using:

\[V = \frac{4}{3} \pi r^3\]

where:

r = radius

V = volume

Examples

First import the module steadystate

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import steadystate as ss

>>> shellLead = ss.Sphere(inner_radius=0.25, outer_radius=0.30, thermalconductivity=35.3)
# This will create an instance of 'Sphere' with a name 'shellLead'

>>> shellLead.volume(radius=shellLead.outer_radius)
0.11309733552923253
# This computes volume @ outer radius

>>> shellLead.volume(radius=shellLead.inner_radius)
0.06544984694978735
# This computes volume @ inner radius

>>>  shellLead.volume(radius=0.25)
0.06544984694978735
# This will also compute volume @ inner radius, which is = 0.25.
# Here the radius is entered as a number directly, rather than as an attribute.